The Spanish international has a year left on his current deal at the London Stadium.
The 31-year-old Adrian arrived at the Hammers in 2013 from Real Betis and has gone on to play 145 times in all competitions for the club.
While he has been brilliant for the club at times, his time at the club has been filled with inconsistency and he has been in and out of the first-choice goalkeeping position.
The Hammers need to try and keep Adrian for at least another season.
Joe Hart is expected to be sent back to Manchester City. The former England first-choice goalkeeper was not good during his loan spell with the club last season and he is not worth the high wages that the club has been paying to keep him at the club.
After taking over as first-choice after Hart was dropped, Adrian delivered decent performances.
While the Hammers will need to sign a new goalkeeper this summer, Hart’s departure means that Adrian could be the only experienced keeper within their squad by the time pre-season starts.
It is important that they have an experienced keeper within the side to add depth and stability to their goalkeeping department ahead of next season.